Problem: Multiply and simplify the following complex numbers: $({-2+4i}) \cdot ({5+i})$
Solution: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({-2+4i}) \cdot ({5+i}) = $ $ ({-2} \cdot {5}) + ({-2} \cdot {i}) + ({4i} \cdot {5}) + ({4i} \cdot {i}) $ Then simplify the terms: $ (-10) + (-2i) + (20i) + (4i^2) $ Imaginary unit multiples can be grouped together. $ -10 + (-2 + 20)i + 4 i^2 $ After we plug in $i^2 = -1$, the result becomes $ -10 + (-2 + 20)i - 4 $ The result is simplified: $ (-10 - 4) + (18i) = -14+18i $